李贵阳外文翻译

IIR Digital Filter Design Gurvinder S Baicher Department of Engineering Newport Business School University of Wales Newport Allt yr yn Campus P O Box 180 Newport NP20 5XR South Wales U K g singh baicher newport ac uk An important step in the development of a digital filter is the determination of a realizable transfer function G z approximating the given frequency response specifications If an IIR filter is desired it is also necessary to ensure that G z is stable The process of deriving the transfer function G z is called digital filter design After G z has been obtained the next step is to realize it in the form of a suitable filter structure In chapter 8 we outlined a variety of basic structures for the realization of FIR and IIR transfer functions In this chapter we consider the IIR digital filter design problem The design of FIR digital filters is treated in chapter 10 First we review some of the issues associated with the filter design problem A widely used approach to IIR filter design based on the conversion of a prototype analog transfer function to a digital transfer function is discussed next Typical design examples are included to illustrate this approach We then consider the transformation of one type of IIR filter transfer function into another type which is achieved by replacing the complex variable z by a function of z Four commonly used transformations are summarized Finally we consider the computer aided design of IIR digital filter To this end we restrict our discussion to the use of matlab in determining the transfer functions 9 1 preliminary considerations There are two major issues that need to be answered before one can develop the digital transfer function G z The first and foremost issue is the development of a reasonable filter frequency response specification from the requirements of the overall system in which the digital filter is to be employed The second issue is to determine whether an FIR or IIR digital filter is to be designed In the section we examine these two issues first Next we review the basic analytical approach to the design of IIR digital filters and then consider the determination of the filter order that meets the prescribed specifications We also discuss appropriate scaling of the transfer function 9 1 1 Digital Filter Specifications As in the case of the analog filter either the magnitude and or the phase delay response is specified for the design of a digital filter for most applications In some situations the unit sample response or step response may be specified In most practical applications the problem of interest is the development of a realizable approximation to a given magnitude response specification As indicated in section 4 6 3 the phase response of the designed filter can be corrected by cascading it with an all pass section The design of all pass phase equalizers has received a fair amount of attention in the last few years We restrict our attention in this chapter to the magnitude approximation problem only We pointed out in section 4 4 1 that there are four basic types of filters whose magnitude responses are shown in Figure 4 10 Since the impulse response corresponding to each of these is no causal and of infinite length these ideal filters are not realizable One way of developing a realizable approximation to these filter would be to truncate the impulse response as indicated in Eq 4 72 for a low pass filter The magnitude response of the FIR low pass filter obtained by truncating the impulse response of the ideal low pass filter does not have a sharp transition from pass band to stop band but rather exhibits a gradual roll off Thus as in the case of the analog filter design problem outlined in section 5 4 1 the magnitude response specifications of a digital filter in the pass band and in the stop band are given with some acceptable tolerances In addition a transition band is specified between the pass band and the stop band to permit the magnitude to drop off smoothly For example the magnitude of a low pass filter may be given j eG as shown in Figure 7 1 As indicated in the figure in the pass band defined by 0 we require that the magnitude approximates unity with an error p of i e In the stop band defined by p pp j p foreG 1 1 we require that the magnitude approximates zero with an error of s e i s for s j eG s The frequencies and are respectively called the pass band edge frequency p s and the stop band edge frequency The limits of the tolerances in the pass band and stop band and are usually called the peak ripple values Note that the p s frequency response of a digital filter is a periodic function of and the j eG magnitude response of a real coefficient digital filter is an even function of As a result the digital filter specifications are given only for the range 0 Digital filter specifications are often given in terms of the loss function in dB Here the peak pass band ripple and the log20 10 j eG p minimum stop band attenuation are given in dB the loss specifications of a s digital filter are given by dB pp 1 log20 10 dB ss log20 10 9 1 Preliminary Considerations As in the case of an analog low pass filter the specifications for a digital low pass filter may alternatively be given in terms of its magnitude response as in Figure 7 2 Here the maximum value of the magnitude in the pass band is assumed to be unity and the maximum pass band deviation denoted as 1 is given by the 2 1 minimum value of the magnitude in the pass band The maximum stop band magnitude is denoted by 1 A For the normalized specification the maximum value of the gain function or the minimum value of the loss function is therefore 0 dB The quantity given by max dB 1 log20 2 10max It is called the maximum pass band attenuation For 1 as is typically the p case it can be shown that pp 2 21 log20 10max The pass band and stop band edge frequencies in most applications are specified in Hz along with the sampling rate of the digital filter Since all filter design techniques are developed in terms of normalized angular frequencies and the p s critical frequencies need to be normalized before a specific filter design algorithm can be applied Let denote the sampling frequency in Hz and FP and Fs denote T F respectively the pass band and stop band edge frequencies in Hz Then the normalized angular edge frequencies in radians are given by TF F F F p T p T p p 2 2 TF F F F s T s T s s 2 2 9 1 2 Selection of the Filter Type The second issue of interest is the selection of the digital filter whether an IIR or an FIR digital filter is to be employed The objective of digital filter design is to develop a causal transfer function H z meeting the frequency response specifications For IIR digital filter design the IIR transfer function is a real rational function of H z 1 z N M dNzzdzdd pMzzpzpp 2 2 1 10 2 2 1 10 Moreover H z must be a stable transfer function and for reduced computational complexity it must be of lowest order N On the other hand for FIR filter design the FIR transfer function is a polynomial in 1 z N n n znhzH 0 For reduced computational complexity the degree N of H z must be as small as possible In addition if a linear phase is desired then the FIR filter coefficients must satisfy the constraint Nnhnh There are several advantages in using an FIR filter since it can be designed with exact linear phase and the filter structure is always stable with quantized filter coefficients However in most cases the order NFIR of an FIR filter is considerably higher than the order NIIR of an equivalent IIR filter meeting the same magnitude specifications In general the implementation of the FIR filter requires approximately NFIR multiplications per output sample whereas the IIR filter requires 2NIIR 1 multiplications per output sample In the former case if the FIR filter is designed with a linear phase then the number of multiplications per output sample reduces to approximately NFIR 1 2 Likewise most IIR filter designs result in transfer functions with zeros on the unit circle and the cascade realization of an IIR filter of order with all of the zeros on the unit circle requires 3 3 2 IIR N IIR N multiplications per output sample It has been shown that for most practical filter specifications the ratio NFIR NIIR is typically of the order of tens or more and as a result the IIR filter usually is computationally more efficient Rab75 However if the group delay of the IIR filter is equalized by cascading it with an all pass equalizer then the savings in computation may no longer be that significant Rab75 In many applications the linearity of the phase response of the digital filter is not an issue is making the IIR filter preferable because of the lower computational requirements 9 1 3 Basic Approaches to Digital Filter Design In the case of IIR filter design the most common practice is to convert the digital filter specifications into analog low pass prototype filter specifications and then to transform it into the desired digital filter transfer function G z This approach has been widely used for many reasons a Analog approximation techniques are highly advanced b They usually yield closed form solutions c Extensive tables are available for analog filter design d Many applications require the digital simulation of analog filters In the sequel we denote an analog transfer function as Where sD sP sH a a a the subscript a specifically indicates the analog domain The digital transfer function derived form Ha s is denoted by zD zP zG The basic idea behind the conversion of an analog prototype transfer function Ha s into a digital IIR transfer function G z is to apply a mapping from the s domain to the z domain so that the essential properties of the analog frequency response are preserved The implies that the mapping function should be such that a The imaginary j axis in the s plane be mapped onto the circle of the z plane b A stable analog transfer function be transformed into a stable digital transfer function To the end the most widely used transformation is the bilinear transformation described in Section 9 2 Unlike IIR digital filter designation The FIR filter design does not have any connection with the design of analog filters The design of FIR filter design does not have any connection with the design of analog filters The design of FIR filters is therefore based on a direct approximation of the specified magnitude responsibility with the often added requirement that the phase response be linear As pointed out in Eq 7 10 a causal FIR transfer function H z of length N 1 is a polynomial in z 1 of degree N The corresponding frequency response is given by N n njj enheH 0 It has been shown in Section 3 2 1 that any finite duration sequence x n of length N 1 is completely characterized by N 1 samples of its discrete time Fourier transfer X As a result the design of an FIR filter of length N 1 may be j e accomplished by finding either the impulse response sequence h n or N 1 samples of its frequency response Also to ensure a linear phase design the H ej condition of Eq 7 11 must be satisfied Two direct approaches to the design of FIR filters are the windowed Fourier series approach and the frequency sampling approach We describe the former approach in Section 7 6 The second approach is treated in Problem 7 6 In Section 7 7 we outline computer based digital filter design methods 中文翻译 中文翻译 IIR 数字滤波器的设计数字滤波器的设计 在一个数字滤波器发展的重要步骤是可实现的传递函数 G z 的接近给定 的频率响应规格 如果一个 IIR 滤波器是理想 它也有必要确保了 G z 是稳 定的 该推算传递函数 G z 的过程称为数字滤波器的设计 然后 G z 有 所值 下一步就是实现在一个合适的过滤器结构形式 在第 8 章 我们概述了 为转移的 FIR 和 IIR 的各种功能的实现基本结构 在这一章中 我们考虑的 IIR 数字滤波器的设计问题 首先 我们回顾与滤波器设计问题相关的一些问题 一种广泛使用的 方法来设计 IIR 滤波器的基础上 传递函数原型模拟到数字的转换传递函数进 行了讨论下一步 典型的设计实例来说明这种方法 然后 我们考虑到另一种 类型 它是由一个函数代替复杂的变量 z 达到了一个 IIR 滤波器的传递函数 z 的类型转换四种常用的转换进行了总结 最后 我们考虑的 IIR 计算机辅助设 计数字滤波器 为此 我们限制我们讨论了 MATLAB 在确定传递函数的使用 9 1 初步考虑 有两个需要先有一个回答可以发展数字传递函数 G z 的重大问题 首要 的问题是一个合理的滤波器的频率响应规格从整个系统中数字滤波器将被雇用 的要求发展 第二个问题是要确定的 FIR 或 IIR 数字滤波器是设计 在一节中 我们首先检查了这两个问题 接下来 我们回顾到的 IIR 数字滤波器设计的基 本分析方法 然后再考虑过滤器的顺序符合规定的规格测定 我们还讨论了传 递函数适当的调整 9 1 1 数字过滤器的规格 如过滤器的模拟案件 无论是规模和 或相位 延迟 响应对于大多数应用程序 指定一个数字滤波器设计 在某些情况下 单位采样响应或阶跃响应可能被指 定 在大多数实际应用中 利益问题是一个变现逼近一个给定的幅度响应的规 范发展 所设计的滤波器可以通过级联与全通区段纠正相位响应 全通相位均 衡器的设计接受了最近几年 相当数量的关注 我们在这方面限制的幅度逼近问题是唯一的一章我们的注意 我们指出 在第 4 4 1 节指出 有四个过滤器 其大小 如图 4 10 所示的反应基本类型 由于 脉冲响应对应于所有这些都是非因果和无限长 这些过滤器是尚未实现的理想 一个发展一个变现的近似值 这些过滤器的方法是截断的脉冲响应 如式所示 4 72 为低通滤波器 该 FIR 低幅度响应滤波器得到截断的理想低通滤波器 从没有一个通带过渡到阻带尖脉冲响应 而是呈现出逐步 下降 因此 正如在模拟滤波器设计 5 4 1 节中所述的问题情况下 在通带数字滤 波器和阻带幅频响应规格给予一些可接受的公差 此外 指定一个过渡带之间 的通带和阻带允许的幅度下降顺利 例如 一个低通滤波器的幅度可能得到如 图 7 1 所示 正如在图中定义的通带 0 我们要求的幅度接近同一个 p 即错误的团结在界定的阻带 我们要求的 pp j p foreG 1 1 幅度接近零与一的错误频率 并分别被称为通带边缘频率和阻带边缘频率 在 通带和阻带 并且误差的限制 通常称为峰值纹波值 请注意 数字滤波器的 频率响应是周期函数 以及幅度响应的实时数字滤波器系数是一个偶函数的 因此 数字滤波规格只给出了范围 数字滤波器的规格 常常给在功能上的损失分贝 在这里 通带纹波和峰值最 小阻带衰减给出了分贝 也就是说 数字滤波器 给出的损失规格 dB pp 1 log20 10 dB ss log20 10 9 1初步设想 正如在一个模拟低通滤波器的情况下 一个数字低通滤波器的规格可能或者给 予其规模在反应方面 如图7 2 在这里 在通带内规模最大的价值被假定为集 中 最大通带偏差 表示为1 是由通带中的最低值所规模 阻带的最 2 1 大震级是指由1 A 对于标准化规格 增益功能或损失函数的最小值最大值 因 此0分贝 给予的数量dB 1 log20 2 10max 被称为最大通带衰减 1 由于通常情况下 它可以证明 p pp 2 21 log20 10max 通带和阻带边缘频率在大多数应用中 被指定为 Hz 随着数字滤波器的采样率 由于所有的过滤器设计技术的规范化发展和角频率来看 临界频率的采样率之 前需要一个特定的过滤器设计算法可以应用于正常化 让表示 在赫兹采样频 率 计划生育和 Fs 分别表示 在通带和阻带的边缘在赫兹频率 然后正常化弧 度角频率都是通过边TF F F F p T p T p p 2 2 TF F F F s T s T s s 2 2 9 1 2过滤器类型的选择 感兴趣的第二个问题是数字滤波器的类型 即选择 无论是原居民或 FIR 数字 滤波器将被雇用 数字滤波器的设计目标是建立一个因果传递函数 H z 的频 率响应规格会议 对于 IIR 数字滤波器的设计 即原传递函数